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Implementation of Oblivious Transfer over Binary-Input AWGN Channels by Polar Codes

Pin-Hsun Lin, Hadi Aghaee, Christian Deppe, Eduard A. Jorswieck, Holger Boche

TL;DR

The paper tackles oblivious transfer over binary-input BI--AWGN channels by marrying polar codes with automorphism-based two-view secrecy. It introduces a structured two-view protocol in which public encodings are randomized over the automorphism group to achieve perfect receiver privacy at finite blocklength, while sender privacy emerges asymptotically through polarization and privacy amplification. A complete algebraic characterization shows that polar-transform automorphisms correspond to bit-position permutations with size $m!$, enabling a finite-blocklength OT-rate optimization that jointly selects the automorphism and the polarized index sets. Reliability is balanced by injecting randomness on selected bad bit-channels and validating performance via Monte Carlo bounds, while a GA-based rate design yields implementable OT payloads on BI--AWGN channels. The result is a PHY-layer OT framework that is both theoretically sound and practically actionable, with explicit procedures for rate optimization and finite-length performance guarantees.

Abstract

We develop a one-out-of-two-oblivious transfer protocol over the binary-input additive white Gaussian noise channel using polar codes. The scheme uses two decoder views linked by automorphisms of the polar transform and publicly draws the encoder at random from the corresponding automorphism group. This yields perfect receiver privacy at any finite blocklength, since the public encoder distribution is independent of the receiver's choice bit. Sender privacy is obtained asymptotically via channel polarization combined with privacy amplification. Because the construction deliberately injects randomness on selected bad bit-channels, we derive a relaxed reliability criterion and evaluate finite-blocklength performance. Finally, we characterize the polar-transform automorphisms as bit-level permutations of bit-channel indices, and exploit this structure to derive and optimize an achievable finite-blocklength OT rate.

Implementation of Oblivious Transfer over Binary-Input AWGN Channels by Polar Codes

TL;DR

The paper tackles oblivious transfer over binary-input BI--AWGN channels by marrying polar codes with automorphism-based two-view secrecy. It introduces a structured two-view protocol in which public encodings are randomized over the automorphism group to achieve perfect receiver privacy at finite blocklength, while sender privacy emerges asymptotically through polarization and privacy amplification. A complete algebraic characterization shows that polar-transform automorphisms correspond to bit-position permutations with size , enabling a finite-blocklength OT-rate optimization that jointly selects the automorphism and the polarized index sets. Reliability is balanced by injecting randomness on selected bad bit-channels and validating performance via Monte Carlo bounds, while a GA-based rate design yields implementable OT payloads on BI--AWGN channels. The result is a PHY-layer OT framework that is both theoretically sound and practically actionable, with explicit procedures for rate optimization and finite-length performance guarantees.

Abstract

We develop a one-out-of-two-oblivious transfer protocol over the binary-input additive white Gaussian noise channel using polar codes. The scheme uses two decoder views linked by automorphisms of the polar transform and publicly draws the encoder at random from the corresponding automorphism group. This yields perfect receiver privacy at any finite blocklength, since the public encoder distribution is independent of the receiver's choice bit. Sender privacy is obtained asymptotically via channel polarization combined with privacy amplification. Because the construction deliberately injects randomness on selected bad bit-channels, we derive a relaxed reliability criterion and evaluate finite-blocklength performance. Finally, we characterize the polar-transform automorphisms as bit-level permutations of bit-channel indices, and exploit this structure to derive and optimize an achievable finite-blocklength OT rate.
Paper Structure (33 sections, 24 theorems, 220 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 33 sections, 24 theorems, 220 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and system model
  3. Notation
  4. Permutation and automorphism
  5. Universal Hash Family
  6. 2-1 OT system setup
  7. Main results
  8. Motivation and setup
  9. The proposed protocol
  10. Impact of parallel decodings at Bob
  11. Baseline decoding: DEC decodes w.r.t. the code $\mathbf{T}$, and $\mathbf P_1=\mathbf P_2\in \mathop{\mathrm{Aut}}\nolimits(\mathbf{T})$
  12. Case 1. DEC decodes w.r.t. $\mathbf T$, but $\mathbf P_1\neq \mathbf P_2$
  13. Case 2. DEC decodes w.r.t. the code $\mathbf P_1^T\mathbf{T}\mathbf P_2$ as a matched decoder
  14. Reliability
  15. and
  16. ...and 18 more sections

Key Result

Theorem 1

If $i \preceq j$, then for every channel $W$ and $n=2^m$, $I\!(W_{n}^{(i)}) \le I\!(W_{n}^{(j)}) \text{ and } Z\!(W_{n}^{(i)}) \ge Z\!(W_{n}^{(j)}).$

Figures (3)

  • Figure 1: BER comparison of letting carry random bits unknown to Bob.
  • Figure 2: The proposed OT system
  • Figure 3: Let $\mathbf{T}_1$-=(11,9) and $\mathbf{T}_1$-=(7,5). Select $\mathbf A\in\mathop{\mathrm{Aut}}\nolimits(\mathbf{T}_1)$, let $\mathbf{T}_2:=\mathbf A\mathbf{T}_1$, such that $\mathbf{T}_2$-=(7,5) and $\mathbf{T}_2$-=(11,9). Let $\mathbf P_1=\mathbf P_2$.

Theorems & Definitions (61)

  • Definition 1
  • Theorem 1: , UPO_Schurch_ISIT16
  • Definition 2
  • Definition 3: Automorphism
  • Definition 4: Cross-cut
  • Lemma 1
  • Definition 5: Partially ordered set
  • Definition 6
  • Example 1
  • Definition 7: Group isomorphism DummitFoote2004
  • ...and 51 more